Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains

Internal gravity wavetrains in continuously stratified fluids are generally unstable as a result of resonant triad interactions which, in the inviscid limit, amplify short-scale perturbations with frequency equal to one half of that of the underlying wave. This so-called parametric subharmonic instability (PSI) has been studied extensively for spatially and temporally monochromatic waves. Here, an asymptotic analysis of PSI for time-harmonic plane waves with locally confined spatial profile is made, in an effort to understand how such wave beams differ, in regard to PSI, from monochromatic plane waves. The discussion centres upon a system of coupled evolution equations that govern the interaction of a small-amplitude wave beam with short-scale subharmonic wavepackets in a nearly inviscid uniformly stratified Boussinesq fluid. For beams with general localized profile, it is found that triad interactions are not strong enough to bring about instability in the limited time that subharmonic perturbations overlap with the beam. On the other hand, for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope, PSI is possible if the beam is wide enough. In this instance, a stability criterion is proposed which, under given flow conditions, provides the minimum number of carrier wavelengths a beam of small amplitude must comprise for instability to arise.